Date of Award
Doctor of Philosophy (PhD)
Christopher B. Croke
Scattering rigidity of a Riemannian manifold allows one to tell the metric of a manifold with boundary by looking at the directions of geodesics at the boundary. Lens rigidity allows one to tell the metric of a manifold with boundary from the same information plus the length of geodesics. There are a variety of results about lens rigidity but very little is known for scattering rigidity. We will discuss the subtle difference between these two types of rigidities and prove that they are equivalent for two-dimensional simple manifolds with boundaries. In particular, this implies
that two-dimensional simple manifolds (such as the
at disk) are scattering rigid since they are lens/boundary rigid (Pestov--Uhlmann, 2005).
Wen, Haomin, "Scattering and Lens Rigidity" (2014). Publicly Accessible Penn Dissertations. 1498.